Theorem lgsdinn0 13549

Description: Variation on lgsdi 13538 valid for all M , N but only for positive A. (The exact location of the failure of this law is for A = -u 1, M = 0, and some N in which case ( -u 1 /L 0 ) = 1 but ( -u 1 /L N ) = -u 1 when -u 1 is not a quadratic residue mod N.) (Contributed by Mario Carneiro, 28-Apr-2016.)

Assertion

Ref	Expression
lgsdinn0	|- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Proof of Theorem lgsdinn0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5849	. . . . . . . . 9 |- ( x = N -> ( A /L x ) = ( A /L N ) )
2	1	oveq1d 5856	. . . . . . . 8 |- ( x = N -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
3	2	eqeq2d 2177	. . . . . . 7 |- ( x = N -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) ) )
4		sq1 10544	. . . . . . . . . . . . . . . . 17 |- ( 1 ^ 2 ) = 1
5	4	eqeq2i 2176	. . . . . . . . . . . . . . . 16 |- ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> ( A ^ 2 ) = 1 )
6		nn0re 9119	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN0 -> A e. RR )
7		nn0ge0 9135	. . . . . . . . . . . . . . . . . 18 |- ( A e. NN0 -> 0 <_ A )
8		1re 7894	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
9		0le1 8375	. . . . . . . . . . . . . . . . . . 19 |- 0 <_ 1
10		sq11 10523	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ 0 <_ 1 ) ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
11	8, 9, 10	mpanr12 436	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. RR /\ 0 <_ A ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
12	6, 7, 11	syl2anc 409	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
13	12	adantr 274	. . . . . . . . . . . . . . . 16 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = ( 1 ^ 2 ) <-> A = 1 ) )
14	5, 13	bitr3id 193	. . . . . . . . . . . . . . 15 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 <-> A = 1 ) )
15	14	biimpa 294	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A = 1 )
16	15	oveq1d 5856	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = ( 1 /L x ) )
17		1lgs 13544	. . . . . . . . . . . . . 14 |- ( x e. ZZ -> ( 1 /L x ) = 1 )
18	17	ad2antlr 481	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 /L x ) = 1 )
19	16, 18	eqtrd 2198	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L x ) = 1 )
20	19	oveq1d 5856	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( 1 x. ( A /L 0 ) ) )
21		nn0z 9207	. . . . . . . . . . . . . . 15 |- ( A e. NN0 -> A e. ZZ )
22	21	ad2antrr 480	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> A e. ZZ )
23		0z 9198	. . . . . . . . . . . . . 14 |- 0 e. ZZ
24		lgscl 13515	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A /L 0 ) e. ZZ )
25	22, 23, 24	sylancl 410	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. ZZ )
26	25	zcnd 9310	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) e. CC )
27	26	mulid2d 7913	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( 1 x. ( A /L 0 ) ) = ( A /L 0 ) )
28	20, 27	eqtr2d 2199	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) = 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
29		lgscl 13515	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
30	21, 29	sylan 281	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. ZZ )
31	30	zcnd 9310	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L x ) e. CC )
32	31	adantr 274	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L x ) e. CC )
33	32	mul01d 8287	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. 0 ) = 0 )
34	21	adantr 274	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ x e. ZZ ) -> A e. ZZ )
35		lgs0 13514	. . . . . . . . . . . . . 14 |- ( A e. ZZ -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
36	34, 35	syl 14	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = if ( ( A ^ 2 ) = 1 , 1 , 0 ) )
37		ifnefalse 3530	. . . . . . . . . . . . 13 |- ( ( A ^ 2 ) =/= 1 -> if ( ( A ^ 2 ) = 1 , 1 , 0 ) = 0 )
38	36, 37	sylan9eq 2218	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = 0 )
39	38	oveq2d 5857	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L x ) x. 0 ) )
40	33, 39, 38	3eqtr4rd 2209	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ x e. ZZ ) /\ ( A ^ 2 ) =/= 1 ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
41		zsqcl 10521	. . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
42	34, 41	syl 14	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A ^ 2 ) e. ZZ )
43		1z 9213	. . . . . . . . . . . 12 |- 1 e. ZZ
44		zdceq 9262	. . . . . . . . . . . 12 |- ( ( ( A ^ 2 ) e. ZZ /\ 1 e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
45	42, 43, 44	sylancl 410	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ x e. ZZ ) -> DECID ( A ^ 2 ) = 1 )
46		dcne 2346	. . . . . . . . . . 11 |- (DECID ( A ^ 2 ) = 1 <-> ( ( A ^ 2 ) = 1 \/ ( A ^ 2 ) =/= 1 ) )
47	45, 46	sylib 121	. . . . . . . . . 10 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( ( A ^ 2 ) = 1 \/ ( A ^ 2 ) =/= 1 ) )
48	28, 40, 47	mpjaodan 788	. . . . . . . . 9 |- ( ( A e. NN0 /\ x e. ZZ ) -> ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
49	48	ralrimiva 2538	. . . . . . . 8 |- ( A e. NN0 -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
50	49	3ad2ant1 1008	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A. x e. ZZ ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) )
51		simp3 989	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
52	3, 50, 51	rspcdva 2834	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
53	52	adantr 274	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L N ) x. ( A /L 0 ) ) )
54	21	3ad2ant1 1008	. . . . . . . . 9 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> A e. ZZ )
55	54, 23, 24	sylancl 410	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. ZZ )
56	55	zcnd 9310	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) e. CC )
57	56	adantr 274	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) e. CC )
58		lgscl 13515	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
59	54, 51, 58	syl2anc 409	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. ZZ )
60	59	zcnd 9310	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L N ) e. CC )
61	60	adantr 274	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L N ) e. CC )
62	57, 61	mulcomd 7916	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L 0 ) x. ( A /L N ) ) = ( ( A /L N ) x. ( A /L 0 ) ) )
63	53, 62	eqtr4d 2201	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L 0 ) = ( ( A /L 0 ) x. ( A /L N ) ) )
64		oveq1 5848	. . . . . 6 |- ( M = 0 -> ( M x. N ) = ( 0 x. N ) )
65	51	zcnd 9310	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> N e. CC )
66	65	mul02d 8286	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( 0 x. N ) = 0 )
67	64, 66	sylan9eqr 2220	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M x. N ) = 0 )
68	67	oveq2d 5857	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
69		simpr 109	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> M = 0 )
70	69	oveq2d 5857	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L M ) = ( A /L 0 ) )
71	70	oveq1d 5856	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L 0 ) x. ( A /L N ) ) )
72	63, 68, 71	3eqtr4d 2208	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
73		oveq2 5849	. . . . . . . 8 |- ( x = M -> ( A /L x ) = ( A /L M ) )
74	73	oveq1d 5856	. . . . . . 7 |- ( x = M -> ( ( A /L x ) x. ( A /L 0 ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
75	74	eqeq2d 2177	. . . . . 6 |- ( x = M -> ( ( A /L 0 ) = ( ( A /L x ) x. ( A /L 0 ) ) <-> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) ) )
76		simp2 988	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
77	75, 50, 76	rspcdva 2834	. . . . 5 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
78	77	adantr 274	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L 0 ) = ( ( A /L M ) x. ( A /L 0 ) ) )
79		oveq2 5849	. . . . . 6 |- ( N = 0 -> ( M x. N ) = ( M x. 0 ) )
80	76	zcnd 9310	. . . . . . 7 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> M e. CC )
81	80	mul01d 8287	. . . . . 6 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( M x. 0 ) = 0 )
82	79, 81	sylan9eqr 2220	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M x. N ) = 0 )
83	82	oveq2d 5857	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( A /L 0 ) )
84		simpr 109	. . . . . 6 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> N = 0 )
85	84	oveq2d 5857	. . . . 5 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L N ) = ( A /L 0 ) )
86	85	oveq2d 5857	. . . 4 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( ( A /L M ) x. ( A /L N ) ) = ( ( A /L M ) x. ( A /L 0 ) ) )
87	78, 83, 86	3eqtr4d 2208	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
88	72, 87	jaodan 787	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
89		neanior 2422	. . 3 |- ( ( M =/= 0 /\ N =/= 0 ) <-> -. ( M = 0 \/ N = 0 ) )
90		lgsdi 13538	. . . 4 |- ( ( ( A e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
91	21, 90	syl3anl1 1276	. . 3 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ ( M =/= 0 /\ N =/= 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
92	89, 91	sylan2br 286	. 2 |- ( ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )
93		zdceq 9262	. . . . 5 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> DECID M = 0 )
94	76, 23, 93	sylancl 410	. . . 4 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID M = 0 )
95		zdceq 9262	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
96	51, 23, 95	sylancl 410	. . . 4 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID N = 0 )
97		dcor 925	. . . 4 |- (DECID M = 0 -> (DECID N = 0 -> DECID ( M = 0 \/ N = 0 ) ) )
98	94, 96, 97	sylc 62	. . 3 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
99		exmiddc 826	. . 3 |- (DECID ( M = 0 \/ N = 0 ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
100	98, 99	syl 14	. 2 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
101	88, 92, 100	mpjaodan 788	1 |- ( ( A e. NN0 /\ M e. ZZ /\ N e. ZZ ) -> ( A /L ( M x. N ) ) = ( ( A /L M ) x. ( A /L N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   A.wral 2443   ifcif 3519   class class class wbr 3981  (class class class)co 5841   CCcc 7747   RRcr 7748   0cc0 7749   1c1 7750   x. cmul 7754   <_ cle 7930   2c2 8904   NN0cn0 9110   ZZcz 9187   ^cexp 10450   /Lclgs 13498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-xor 1366  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-irdg 6334  df-frec 6355  df-1o 6380  df-2o 6381  df-oadd 6384  df-er 6497  df-en 6703  df-dom 6704  df-fin 6705  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-5 8915  df-6 8916  df-7 8917  df-8 8918  df-9 8919  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-ihash 10685  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-clim 11216  df-proddc 11488  df-dvds 11724  df-gcd 11872  df-prm 12036  df-phi 12139  df-pc 12213  df-lgs 13499

This theorem is referenced by: (None)